iS2 PHILOSOPHICAL TRANSACTIONS. [aNNo1771. 



of the same fluxion generated while z from -^-r-^ becomes = m; cp being 

 taken = n V'-« 



Suppose /J = — ^ — -r, its value will then be - vm' + «^ . Consequently 



the fluent of — =A?!=r^L=-, generated while 2 from O becomes = - v/m^ -4- n^ — 

 — , together with the quantity ^m'-{-n'^ — v, is equal to the fluent of the same 



fluxion generated while z from - ^m^ + n^ becomes = m : and these two 



parts of the whole fluent being denoted by m and n respectively; m will be = 

 n — AE, and n = V m'^ + n^ — ae. _ 



7. The fluent of -—t^^L= being l + ad - dp, the fluent of — J^^?L= 



•v^n* + 2/z - «» _yn' + 9fi - z* 



+ DP — AD — L will be = O. Therefore, the fluent of — ^^ TL. ■ + the 



Vn^ + Ijz - 2« 



fluent of —pJL=~=L being = the fluent of \z^z v/^- — ^-, it is obvious, 

 that the fluent of ~^^~=-^ is = dp — ad — l + the fluent of 4- z"^i x 



■V n» + Zfi — z' 



^ +"'^ _.pp_AP — L + the elliptic arc dg (fig. 8) whose abscissa cp is = ny/-. 

 Consequently, putting e for + of the periphery of that ellipsis, it appears that 



the whole fluent of .- — '— - , generated while z from O becomes = m, is 



V «•' + 2/2 — i" ° ' 



equal to e — l = e + 2ae — n — ^ ni^ + "^• 



8. By taking, in art. 3, q, r, and s, each = 4-; and a = — d = 



—.6=1, and c = n-"; we find, that, if w be = — , , — , _- —=-M——-^ -" 



»» ' ' ' ' .y »' + OTZ ' Vn« + 2/2—2'^ -/»= + 2/j/-y' 



will be = 0. 



It is obvious therefore, that the fluent of -r:= — ■- , generated while z 



A n' + 2/z — 2« *' 



from O becomes equal to any quantity k, is equal to the fluent of the same 

 fluxion, generated while z from " ~" becomes = m. 



Now, supposing k = — ^ — T-, its value will be - v' »»'' + n^ . 



Consequently, the fluent of 



•~7=^ — ^^^^^> generated while z from becomes = - V jn^ -}- ri' — ~, is equal 



to half the fluent of the same fluxion, generated while z from O becomes = m; 

 which half fluent is known by the preceding article. 



0. It appears, by art. 4, that -7 — - -^= + — r —'=■ is = — the flux. 



" rr ' .; V„j _j. yy _ yx v„. + 2/z — z» 



of the tang, dt; and it appears, by the last article, that 



i«'iVi , Jn'zv'i . , , , , . 



-z=^ — ==^=- + —p ^^^— IS = O; mn' — n y — nz — mxiz bemff =. O. 



